Construction of matryoshka nested indecomposable N-replications of Kac-modules of quasi-reductive Lie superalgebras, including the sl(m/n) and osp(2/2n) series

TL;DR

This paper introduces a novel method to construct nested indecomposable representations of certain Lie superalgebras, providing a potential mathematical framework for understanding particle generations in physics.

Contribution

It presents a recursive construction of N-replications of Kac-modules in quasi-reductive Lie superalgebras, extending previous representation theory.

Findings

Constructed finite dimensional indecomposable representations of superalgebras.

Demonstrated the recursive embedding of multiple copies of irreducible representations.

Connected the mathematical construction to particle generation phenomena.

Abstract

We construct a new class of finite dimensional indecomposable representations of simple superalgebras which may explain, in a natural way, the existence of the heavier elementary particles. In type I Lie superalgebras sl(m/n) and osp(2/2n), one of the Dynkin weights labeling the finite dimensional irreducible representations is continuous. Taking the derivative, we show how to construct indecomposable representations recursively embedding N copies of the original irreducible representation, coupled by generalized Cabibbo angles, as observed among the three generations of leptons and quarks of the standard model. The construction is then generalized in the appendix to quasi-reductive Lie superalgebras.